The poinr of suspension of a plane simple pendulum of mass m and length I is con

Question

The poinr of suspension of a plane simple pendulum of mass m and length I is constrained 10 move along a horizontal track and is connected to a point on the circumference of a uniform flywheel of mass M and radius a through a mass-less connecting rod also of length a, as shown in die figure. The flywheel rotates about a center fixed on the track. Find a Hamiltonian for the combined system and determine Hamilton's equations of motion. Suppose die po.m of suspension were moved along the track accoiding to some function of time x = f(t), where x reverses at x = plusminus 2a (relative to the center of the fly wheel). Again, find a Hamiltonian and Hamilton's equations of motion.

Explanation / Answer

A simple pendulum is an idealized model consisting of a point mass suspended by a massless,unstretchable string.When the point mass is pulled to one side of its straight-down equilibrium position and released,it oscillates about the equilibrium position. We represent the forces on the mass in terms of tangential and radial components.The restoring force F? is the tangential component of the net force: F? = -mg * sin? -----------(1) The restoring force is provided by gravity;the tension T merely acts to make the point mass move in an arc.The restoring force is proportional not to ? but to sin?,so the motion is not simple harmonic.With this approximation,equation (1) becomes F? = -mg? = -mg(x/L) or F? = -(mg/L) * x The restoring force is then proportional to the coordinate for small displacements,and the force constant is k = (mg/L).From equation (1) the angular frequency w of a simple pendulum with small amplitude is w = (k/m)^1/2 = ((mg/L)/m)^1/2 = (g/L)^1/2 (simple pendulum,small amplitude) The corresponding frequency and period relations are f = (w/2p) = (1/2p) * (g/L)^1/2 or T = (2p/w) = (1/f) = 2p * (L/g)^1/2 (simple pendulum,small amplitude) Note that these expressions do not involve the mass of the particle.This is because the restoring force,a component of the partilce's weight,is proportional to m.

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